I replaced some existing incandescent bulbs by LED bulbs, then noticed
that they do not switch off completely but still glow dimly.

Checking the voltage at the lamp socket with a high-resistance
electronic multimeter, I get a reading of 48v!! A few volts might be OK
-- induction, stray voltages, etc., but 48V!!!

This occurs in two different rooms, but I think the various lighting
circuits are all connected to one breaker.

What could cause this?

Perce